Power and Sample Size for some Chi-Square Goodness of Fit Tests

Abstract

The first decision a researcher must make is to decide what sample size will be used in the experiment. Many researchers are familiar with sample size issues for the simple t-test, approximate binomial tests, two-sample t-test, and the analysis of variance. However, it is very difficult to find anyone who is familiar with the power and sample size issues for the Chi-Square goodness of fit test. For example, the first hypothesis examined is testing to see if two binomial proportions are equal. An approximate test of this hypothesis can be conducted using either a z-test or a Chi-Square goodness of fit test. These tests are equivalent tests since z^2= χ^2. The power of these tests can be approximated by using the standard normal distribution or a non-central Chi-Square distribution. I derived the non-centrality parameter for this test and the other related goodness of fit tests. It is somewhat surprising that the power of the test computed using the standard normal is not identical to the power based on the Chi-Square. Even though the values are not equal they are quite close. A simulation is also conducted to estimate alpha and power. The results show that the empirical level of significance for the Chi-Square goodness test is close to alpha. It is also seen that simulated powers tend to be quite close to powers computed using the non-central Chi-Square. Some simple iterative programs are included that can be used to compute the sample size needed to detect a given departure from the null hypothesis with a desired power.